Simplify; express your answer in exponential form. Assume $a\neq 0, p\neq 0$. $\dfrac{{(a^{-1})^{-1}}}{{(a^{4}p^{-5})^{2}}}$
Solution: To start, try working on the numerator and the denominator independently. In the numerator, we have ${a^{-1}}$ to the exponent ${-1}$ . Now ${-1 \times -1 = 1}$ , so ${(a^{-1})^{-1} = a}$ In the denominator, we can use the distributive property of exponents. ${(a^{4}p^{-5})^{2} = (a^{4})^{2}(p^{-5})^{2}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(a^{-1})^{-1}}}{{(a^{4}p^{-5})^{2}}} = \dfrac{{a}}{{a^{8}p^{-10}}}$ Break up the equation by variable and simplify. $\dfrac{{a}}{{a^{8}p^{-10}}} = \dfrac{{a}}{{a^{8}}} \cdot \dfrac{{1}}{{p^{-10}}} = a^{{1} - {8}} \cdot p^{- {(-10)}} = a^{-7}p^{10}$.